I believe you can refuse to comply with those checkpoints.
As an aside, the ACLU's "Constitution free zone" campaign isn't referring to citizenship checks†; it's referring to a DHS employee's insistence that being within N miles of anything that could be considered a border subjects you to warrantless border searches. Which, again, is something you can find a SCOTUS case refuting.
† Unfortunately, it's now apparently the law of the land that the police can demand identification from citizens, which is one of the great BS decisions of the last 20 years.
